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Johnson solid
(J''37), a Johnson solid]] example is not a Johnson solid because it is not convex. (This particular regular polyhedron is actually a stellation, the only possible one for the octahedron.)]] s.)]] In geometry, a '''Johnson solid' is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (''J''1); it has 1 square face and 4 triangular faces. As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid (J''2) is an example that actually has a degree-5 vertex. Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3, 4, 5, 6, 8, or 10 sides. In 1966, Norman Johnson published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete. Of the Johnson solids, the elongated square gyrobicupola (''J''37) is unique in being locally vertex-uniform: there are 4 faces at each vertex, and their arrangement is always the same: 3 squares and 1 triangle. However, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid. Names The names are listed below and are more descriptive than they sound. Most of the Johnson solids can be constructed from the first few (pyramids, cupolae, and rotundae), together with the Platonic and Archimedean solids, prisms, and antiprisms. *''Bi-'' means that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, they can be joined so that like faces (''ortho-'') or unlike faces (''gyro-'') meet. In this nomenclature, an octahedron would be a ''square bipyramid, a cuboctahedron would be a triangular gyrobicupola, and an icosidodecahedron would be a pentagonal gyrobirotunda. *''Elongated'' means that a prism has been joined to the base of the solid in question or between the bases of the solids in question. A rhombicuboctahedron would be an elongated square orthobicupola. *''Gyroelongated'' means that an antiprism has been joined to the base of the solid in question or between the bases of the solids in question. An icosahedron would be a gyroelongated pentagonal bipyramid. *''Augmented'' means that a pyramid or cupola has been joined to a face of the solid in question. *''Diminished'' means that a pyramid or cupola has been removed from the solid in question. *''Gyrate'' means that a cupola on the solid in question has been rotated so that different edges match up, as in the difference between ortho- and gyrobicupolae. The last three operations — augmentation, diminution, and gyration — can be performed more than once on a large enough solid. We add bi-'' to the name of the operation to indicate that it has been performed twice. (A ''bigyrate solid has had two of its cupolae rotated.) We add tri-'' to indicate that it has been performed three times. (A ''tridiminished solid has had three of its pyramids or cupolae removed.) Sometimes, bi-'' alone is not specific enough. We must distinguish between a solid that has had two parallel faces altered and one that has had two oblique faces altered. When the faces altered are parallel, we add ''para-'' to the name of the operation. (A ''parabiaugmented solid has had two parallel faces augmented.) When they are not, we add meta-'' to the name of the operation. (A ''metabiaugmented solid has had 2 oblique faces augmented.) Enumeration Prismatoids and rotundae * Pyramids * Cupolas * Rotunda Modified pyramids and dipyramids *elongated pyramid *gyroelongated pyramid *bipyramid *elongated dipyramid *gyroelongated dipyramid Modified cupolas and rotunda * elongated cupola * elongated rotunda * elongated birotunda * elongated cupolarotunda * elongated bicupola * gyroelongated cupola * gyroelongated rotunda * bicupola * cupolarotunda * gyroelongated bicupola * gyroelongated birotunda * gyroelongated cupolarotunda Augmented prisms Modified Platonic solids * Augmented dodecahedrons * Diminished icosahedrons Modified Archimedean solids * augmented truncated tetrahedron * augmented truncated cube * augmented truncated dodecahedron * gyrate rhombicosadodecahedron * diminished rhombicosadodecahedron * gyrate diminished rhombicosadodecahedron * diminished rhombicosadodecahedron * gyrate diminished rhombicosadodecahedron * diminished rhombicosadodecahedron Miscellaneous See also * Near-miss Johnson solid References *Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others. * The first proof that there are only 92 Johnson solids. External links * Sylvain Gagnon, "Convex polyhedra with regular faces", Structural Topology, No. 6, 1982, 83-95. *Paper Models of Polyhedra Many links *Johnson Solids by George W. Hart. *Images of all 92 solids, categorized, on one page * *VRML models *VRML models of Johnson Solids by Vladimir Bulatov * Category:Polyhedra ca:Políedre de Johnson de:Johnson-Körper eo:Solido de Johnson es:Sólido de Johnson fr:Solide de Johnson it:Solido di Johnson nl:Johnson-lichaam ja:ジョンソンの立体 pt:Sólidos de Johnson th:ทรงตันจอห์นสัน zh:约翰逊多面体